$T:P_{3}\rightarrow P_{3}$ defined by $T(p(t))=tp'(t)+p(0)$ is a linear transformation.

Determine whether $T$ is invertible.

If yes, find $T^{-1}(q(t))$, where $q(t)$ is a polynomial of degree at most three.

One last thing... I have got that if $p(t)=at^3+bt^2+c^t+d$, then $T(p(t))=3at^3+2bt^2+c^t+d$. I understand that I can write the inverse of T as a matrix $[\frac{1}{3}, \frac{1}{2}, 1, 1]$, but how to put that into a nice form like the one $p(t)$ is in?